Here are some common (and less common) parent functions:
name
\(f(x)=\)
constant
1
linear
\(x\)
absolute
\(|x|\)
quadratic
\(x^2\)
cubic
\(x^3\)
reciprocal
\(\frac{1}{x}\)
square root
\(\sqrt{x}\)
cube root
\(\sqrt[3]{x}\)
sine
\(\sin(x)\)
cosine
\(\cos(x)\)
tangent
\(\tan(x)\)
ceiling
\(\lceil x \rceil\)
floor
\(\lfloor x \rfloor\)
exponential
\(e^x\)
logarithmic
\(\ln(x)\)
logistic
\(\frac{e^x}{e^x+1}\)
squared reciprocal
\(\frac{1}{x^2}\)
Match the graphs with their names.
Solution
You could try graphing the options or referencing a parent-function chart.
Question
Here are some common (and less common) parent functions:
name
\(f(x)=\)
constant
1
linear
\(x\)
absolute
\(|x|\)
quadratic
\(x^2\)
cubic
\(x^3\)
reciprocal
\(\frac{1}{x}\)
square root
\(\sqrt{x}\)
cube root
\(\sqrt[3]{x}\)
sine
\(\sin(x)\)
cosine
\(\cos(x)\)
tangent
\(\tan(x)\)
ceiling
\(\lceil x \rceil\)
floor
\(\lfloor x \rfloor\)
exponential
\(e^x\)
logarithmic
\(\ln(x)\)
logistic
\(\frac{e^x}{e^x+1}\)
squared reciprocal
\(\frac{1}{x^2}\)
Match the graphs with their names.
Solution
You could try graphing the options or referencing a parent-function chart.
Question
Here are some common (and less common) parent functions:
name
\(f(x)=\)
constant
1
linear
\(x\)
absolute
\(|x|\)
quadratic
\(x^2\)
cubic
\(x^3\)
reciprocal
\(\frac{1}{x}\)
square root
\(\sqrt{x}\)
cube root
\(\sqrt[3]{x}\)
sine
\(\sin(x)\)
cosine
\(\cos(x)\)
tangent
\(\tan(x)\)
ceiling
\(\lceil x \rceil\)
floor
\(\lfloor x \rfloor\)
exponential
\(e^x\)
logarithmic
\(\ln(x)\)
logistic
\(\frac{e^x}{e^x+1}\)
squared reciprocal
\(\frac{1}{x^2}\)
Match the graphs with their names.
Solution
You could try graphing the options or referencing a parent-function chart.
Question
Here are some common (and less common) parent functions:
name
\(f(x)=\)
constant
1
linear
\(x\)
absolute
\(|x|\)
quadratic
\(x^2\)
cubic
\(x^3\)
reciprocal
\(\frac{1}{x}\)
square root
\(\sqrt{x}\)
cube root
\(\sqrt[3]{x}\)
sine
\(\sin(x)\)
cosine
\(\cos(x)\)
tangent
\(\tan(x)\)
ceiling
\(\lceil x \rceil\)
floor
\(\lfloor x \rfloor\)
exponential
\(e^x\)
logarithmic
\(\ln(x)\)
logistic
\(\frac{e^x}{e^x+1}\)
squared reciprocal
\(\frac{1}{x^2}\)
Match the graphs with their names.
Solution
You could try graphing the options or referencing a parent-function chart.
Question
Here are some common (and less common) parent functions:
name
\(f(x)=\)
constant
1
linear
\(x\)
absolute
\(|x|\)
quadratic
\(x^2\)
cubic
\(x^3\)
reciprocal
\(\frac{1}{x}\)
square root
\(\sqrt{x}\)
cube root
\(\sqrt[3]{x}\)
sine
\(\sin(x)\)
cosine
\(\cos(x)\)
tangent
\(\tan(x)\)
ceiling
\(\lceil x \rceil\)
floor
\(\lfloor x \rfloor\)
exponential
\(e^x\)
logarithmic
\(\ln(x)\)
logistic
\(\frac{e^x}{e^x+1}\)
squared reciprocal
\(\frac{1}{x^2}\)
Match the graphs with their names.
Solution
You could try graphing the options or referencing a parent-function chart.
Question
Here are some common (and less common) parent functions:
name
\(f(x)=\)
constant
1
linear
\(x\)
absolute
\(|x|\)
quadratic
\(x^2\)
cubic
\(x^3\)
reciprocal
\(\frac{1}{x}\)
square root
\(\sqrt{x}\)
cube root
\(\sqrt[3]{x}\)
sine
\(\sin(x)\)
cosine
\(\cos(x)\)
tangent
\(\tan(x)\)
ceiling
\(\lceil x \rceil\)
floor
\(\lfloor x \rfloor\)
exponential
\(e^x\)
logarithmic
\(\ln(x)\)
logistic
\(\frac{e^x}{e^x+1}\)
squared reciprocal
\(\frac{1}{x^2}\)
Match the graphs with their names.
Solution
You could try graphing the options or referencing a parent-function chart.
Question
Here are some common (and less common) parent functions:
name
\(f(x)=\)
constant
1
linear
\(x\)
absolute
\(|x|\)
quadratic
\(x^2\)
cubic
\(x^3\)
reciprocal
\(\frac{1}{x}\)
square root
\(\sqrt{x}\)
cube root
\(\sqrt[3]{x}\)
sine
\(\sin(x)\)
cosine
\(\cos(x)\)
tangent
\(\tan(x)\)
ceiling
\(\lceil x \rceil\)
floor
\(\lfloor x \rfloor\)
exponential
\(e^x\)
logarithmic
\(\ln(x)\)
logistic
\(\frac{e^x}{e^x+1}\)
squared reciprocal
\(\frac{1}{x^2}\)
Match the graphs with their names.
Solution
You could try graphing the options or referencing a parent-function chart.
Question
Here are some common (and less common) parent functions:
name
\(f(x)=\)
constant
1
linear
\(x\)
absolute
\(|x|\)
quadratic
\(x^2\)
cubic
\(x^3\)
reciprocal
\(\frac{1}{x}\)
square root
\(\sqrt{x}\)
cube root
\(\sqrt[3]{x}\)
sine
\(\sin(x)\)
cosine
\(\cos(x)\)
tangent
\(\tan(x)\)
ceiling
\(\lceil x \rceil\)
floor
\(\lfloor x \rfloor\)
exponential
\(e^x\)
logarithmic
\(\ln(x)\)
logistic
\(\frac{e^x}{e^x+1}\)
squared reciprocal
\(\frac{1}{x^2}\)
Match the graphs with their names.
Solution
You could try graphing the options or referencing a parent-function chart.
Question
Here are some common (and less common) parent functions:
name
\(f(x)=\)
constant
1
linear
\(x\)
absolute
\(|x|\)
quadratic
\(x^2\)
cubic
\(x^3\)
reciprocal
\(\frac{1}{x}\)
square root
\(\sqrt{x}\)
cube root
\(\sqrt[3]{x}\)
sine
\(\sin(x)\)
cosine
\(\cos(x)\)
tangent
\(\tan(x)\)
ceiling
\(\lceil x \rceil\)
floor
\(\lfloor x \rfloor\)
exponential
\(e^x\)
logarithmic
\(\ln(x)\)
logistic
\(\frac{e^x}{e^x+1}\)
squared reciprocal
\(\frac{1}{x^2}\)
Match the graphs with their names.
Solution
You could try graphing the options or referencing a parent-function chart.
Question
Here are some common (and less common) parent functions:
name
\(f(x)=\)
constant
1
linear
\(x\)
absolute
\(|x|\)
quadratic
\(x^2\)
cubic
\(x^3\)
reciprocal
\(\frac{1}{x}\)
square root
\(\sqrt{x}\)
cube root
\(\sqrt[3]{x}\)
sine
\(\sin(x)\)
cosine
\(\cos(x)\)
tangent
\(\tan(x)\)
ceiling
\(\lceil x \rceil\)
floor
\(\lfloor x \rfloor\)
exponential
\(e^x\)
logarithmic
\(\ln(x)\)
logistic
\(\frac{e^x}{e^x+1}\)
squared reciprocal
\(\frac{1}{x^2}\)
Match the graphs with their names.
Solution
You could try graphing the options or referencing a parent-function chart.
Question
Consider the quadratic function graphed below:
What is the interval(s) over which the function is positive?
What is the interval(s) over which the function is negative?
What is the interval(s) over which the function is increasing?
What is the interval(s) over which the function is decreasing?
What is the domain (in interval notation)?
What is the range (in interval notation)?
Solution
Positive: (-7,1)
Negative: (-inf,-7)U(1,inf)
Increasing: (-inf,-3)
Decreasing: (-3,inf)
Domain: (-inf,inf)
Range: (-inf,7]
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-7,1)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-inf,-7)U(1,inf)
The function is increasing when an ant going left-to-right is walking up hill: (-inf,-3)
The function is decreasing when an ant going left-to-right is walking down hill: (-3,inf)
Question
Consider the quadratic function graphed below:
What is the interval(s) over which the function is positive?
What is the interval(s) over which the function is negative?
What is the interval(s) over which the function is increasing?
What is the interval(s) over which the function is decreasing?
What is the domain (in interval notation)?
What is the range (in interval notation)?
Solution
Positive: (-inf,-8)U(0,inf)
Negative: (-8,0)
Increasing: (-4,inf)
Decreasing: (-inf,-4)
Domain: (-inf,inf)
Range: [-1,inf)
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-inf,-8)U(0,inf)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-8,0)
The function is increasing when an ant going left-to-right is walking up hill: (-4,inf)
The function is decreasing when an ant going left-to-right is walking down hill: (-inf,-4)
Question
Consider the quadratic function graphed below:
What is the interval(s) over which the function is positive?
What is the interval(s) over which the function is negative?
What is the interval(s) over which the function is increasing?
What is the interval(s) over which the function is decreasing?
What is the domain (in interval notation)?
What is the range (in interval notation)?
Solution
Positive: (-inf,-3)U(5,inf)
Negative: (-3,5)
Increasing: (1,inf)
Decreasing: (-inf,1)
Domain: (-inf,inf)
Range: [-2,inf)
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-inf,-3)U(5,inf)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-3,5)
The function is increasing when an ant going left-to-right is walking up hill: (1,inf)
The function is decreasing when an ant going left-to-right is walking down hill: (-inf,1)
Question
Consider the quadratic function graphed below:
What is the interval(s) over which the function is positive?
What is the interval(s) over which the function is negative?
What is the interval(s) over which the function is increasing?
What is the interval(s) over which the function is decreasing?
What is the domain (in interval notation)?
What is the range (in interval notation)?
Solution
Positive: (-5,3)
Negative: (-inf,-5)U(3,inf)
Increasing: (-inf,-1)
Decreasing: (-1,inf)
Domain: (-inf,inf)
Range: (-inf,4]
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-5,3)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-inf,-5)U(3,inf)
The function is increasing when an ant going left-to-right is walking up hill: (-inf,-1)
The function is decreasing when an ant going left-to-right is walking down hill: (-1,inf)
Question
Consider the quadratic function graphed below:
What is the interval(s) over which the function is positive?
What is the interval(s) over which the function is negative?
What is the interval(s) over which the function is increasing?
What is the interval(s) over which the function is decreasing?
What is the domain (in interval notation)?
What is the range (in interval notation)?
Solution
Positive: (-6,-2)
Negative: (-inf,-6)U(-2,inf)
Increasing: (-inf,-4)
Decreasing: (-4,inf)
Domain: (-inf,inf)
Range: (-inf,5]
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-6,-2)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-inf,-6)U(-2,inf)
The function is increasing when an ant going left-to-right is walking up hill: (-inf,-4)
The function is decreasing when an ant going left-to-right is walking down hill: (-4,inf)
Question
Consider the quadratic function graphed below:
What is the interval(s) over which the function is positive?
What is the interval(s) over which the function is negative?
What is the interval(s) over which the function is increasing?
What is the interval(s) over which the function is decreasing?
What is the domain (in interval notation)?
What is the range (in interval notation)?
Solution
Positive: (-inf,-7)U(3,inf)
Negative: (-7,3)
Increasing: (-2,inf)
Decreasing: (-inf,-2)
Domain: (-inf,inf)
Range: [-5,inf)
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-inf,-7)U(3,inf)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-7,3)
The function is increasing when an ant going left-to-right is walking up hill: (-2,inf)
The function is decreasing when an ant going left-to-right is walking down hill: (-inf,-2)
Question
Consider the quadratic function graphed below:
What is the interval(s) over which the function is positive?
What is the interval(s) over which the function is negative?
What is the interval(s) over which the function is increasing?
What is the interval(s) over which the function is decreasing?
What is the domain (in interval notation)?
What is the range (in interval notation)?
Solution
Positive: (-inf,-2)U(8,inf)
Negative: (-2,8)
Increasing: (3,inf)
Decreasing: (-inf,3)
Domain: (-inf,inf)
Range: [-6,inf)
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-inf,-2)U(8,inf)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-2,8)
The function is increasing when an ant going left-to-right is walking up hill: (3,inf)
The function is decreasing when an ant going left-to-right is walking down hill: (-inf,3)
Question
Consider the quadratic function graphed below:
What is the interval(s) over which the function is positive?
What is the interval(s) over which the function is negative?
What is the interval(s) over which the function is increasing?
What is the interval(s) over which the function is decreasing?
What is the domain (in interval notation)?
What is the range (in interval notation)?
Solution
Positive: (-5,5)
Negative: (-inf,-5)U(5,inf)
Increasing: (-inf,0)
Decreasing: (0,inf)
Domain: (-inf,inf)
Range: (-inf,7]
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-5,5)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-inf,-5)U(5,inf)
The function is increasing when an ant going left-to-right is walking up hill: (-inf,0)
The function is decreasing when an ant going left-to-right is walking down hill: (0,inf)
Question
Consider the quadratic function graphed below:
What is the interval(s) over which the function is positive?
What is the interval(s) over which the function is negative?
What is the interval(s) over which the function is increasing?
What is the interval(s) over which the function is decreasing?
What is the domain (in interval notation)?
What is the range (in interval notation)?
Solution
Positive: (-inf,0)U(6,inf)
Negative: (0,6)
Increasing: (3,inf)
Decreasing: (-inf,3)
Domain: (-inf,inf)
Range: [-1,inf)
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-inf,0)U(6,inf)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (0,6)
The function is increasing when an ant going left-to-right is walking up hill: (3,inf)
The function is decreasing when an ant going left-to-right is walking down hill: (-inf,3)
Question
Consider the quadratic function graphed below:
What is the interval(s) over which the function is positive?
What is the interval(s) over which the function is negative?
What is the interval(s) over which the function is increasing?
What is the interval(s) over which the function is decreasing?
What is the domain (in interval notation)?
What is the range (in interval notation)?
Solution
Positive: (-5,3)
Negative: (-inf,-5)U(3,inf)
Increasing: (-inf,-1)
Decreasing: (-1,inf)
Domain: (-inf,inf)
Range: (-inf,7]
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-5,3)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-inf,-5)U(3,inf)
The function is increasing when an ant going left-to-right is walking up hill: (-inf,-1)
The function is decreasing when an ant going left-to-right is walking down hill: (-1,inf)
Question
Which plot matches the function:
\[f(x) = \left|{x - 1}\right| + 6\]
Plot 1
Plot 2
Plot 3
Plot 4
Solution
The parent function is \(f(x)=|x|\), which has a vertex at the origin.
The general parameterization is \(f(x)=a\left|x-h\right|+k\). The vertex shifts to \((h,k)\). The multiplicative factor, \(a\), causes a vertical stretch and/or a vertical flip.
In this example, \(a=1\) and \(h=1\) and \(k=6\).
To get the correct daughter graph, translate the parent 1 units right and 6 units up. Since \(a>0\), the daughter points up.
The correct plot is Plot 4.
\[f(x) = \left|{x - 1}\right| + 6\]
Question
Which plot matches the function:
\[f(x) = \left|{x + 4}\right| - 6\]
Plot 1
Plot 2
Plot 3
Plot 4
Solution
The parent function is \(f(x)=|x|\), which has a vertex at the origin.
The general parameterization is \(f(x)=a\left|x-h\right|+k\). The vertex shifts to \((h,k)\). The multiplicative factor, \(a\), causes a vertical stretch and/or a vertical flip.
In this example, \(a=1\) and \(h=-4\) and \(k=-6\).
To get the correct daughter graph, translate the parent 4 units left and 6 units down. Since \(a>0\), the daughter points up.
The correct plot is Plot 2.
\[f(x) = \left|{x + 4}\right| - 6\]
Question
Which plot matches the function:
\[f(x) = \left|{x + 2}\right| + 1\]
Plot 1
Plot 2
Plot 3
Plot 4
Solution
The parent function is \(f(x)=|x|\), which has a vertex at the origin.
The general parameterization is \(f(x)=a\left|x-h\right|+k\). The vertex shifts to \((h,k)\). The multiplicative factor, \(a\), causes a vertical stretch and/or a vertical flip.
In this example, \(a=1\) and \(h=-2\) and \(k=1\).
To get the correct daughter graph, translate the parent 2 units left and 1 units up. Since \(a>0\), the daughter points up.
The correct plot is Plot 4.
\[f(x) = \left|{x + 2}\right| + 1\]
Question
Which plot matches the function:
\[f(x) = - \left|{x - 3}\right| - 6\]
Plot 1
Plot 2
Plot 3
Plot 4
Solution
The parent function is \(f(x)=|x|\), which has a vertex at the origin.
The general parameterization is \(f(x)=a\left|x-h\right|+k\). The vertex shifts to \((h,k)\). The multiplicative factor, \(a\), causes a vertical stretch and/or a vertical flip.
In this example, \(a=-1\) and \(h=3\) and \(k=-6\).
To get the correct daughter graph, translate the parent 3 units right and 6 units down. Since \(a<0\), the daughter points down.
The correct plot is Plot 3.
\[f(x) = - \left|{x - 3}\right| - 6\]
Question
Which plot matches the function:
\[f(x) = 4 - \left|{x + 5}\right|\]
Plot 1
Plot 2
Plot 3
Plot 4
Solution
The parent function is \(f(x)=|x|\), which has a vertex at the origin.
The general parameterization is \(f(x)=a\left|x-h\right|+k\). The vertex shifts to \((h,k)\). The multiplicative factor, \(a\), causes a vertical stretch and/or a vertical flip.
In this example, \(a=-1\) and \(h=-5\) and \(k=4\).
To get the correct daughter graph, translate the parent 5 units left and 4 units up. Since \(a<0\), the daughter points down.
The correct plot is Plot 3.
\[f(x) = 4 - \left|{x + 5}\right|\]
Question
Which plot matches the function:
\[f(x) = \left|{x + 4}\right| + 5\]
Plot 1
Plot 2
Plot 3
Plot 4
Solution
The parent function is \(f(x)=|x|\), which has a vertex at the origin.
The general parameterization is \(f(x)=a\left|x-h\right|+k\). The vertex shifts to \((h,k)\). The multiplicative factor, \(a\), causes a vertical stretch and/or a vertical flip.
In this example, \(a=1\) and \(h=-4\) and \(k=5\).
To get the correct daughter graph, translate the parent 4 units left and 5 units up. Since \(a>0\), the daughter points up.
The correct plot is Plot 4.
\[f(x) = \left|{x + 4}\right| + 5\]
Question
Which plot matches the function:
\[f(x) = \left|{x - 3}\right| - 4\]
Plot 1
Plot 2
Plot 3
Plot 4
Solution
The parent function is \(f(x)=|x|\), which has a vertex at the origin.
The general parameterization is \(f(x)=a\left|x-h\right|+k\). The vertex shifts to \((h,k)\). The multiplicative factor, \(a\), causes a vertical stretch and/or a vertical flip.
In this example, \(a=1\) and \(h=3\) and \(k=-4\).
To get the correct daughter graph, translate the parent 3 units right and 4 units down. Since \(a>0\), the daughter points up.
The correct plot is Plot 4.
\[f(x) = \left|{x - 3}\right| - 4\]
Question
Which plot matches the function:
\[f(x) = - \left|{x + 6}\right| - 4\]
Plot 1
Plot 2
Plot 3
Plot 4
Solution
The parent function is \(f(x)=|x|\), which has a vertex at the origin.
The general parameterization is \(f(x)=a\left|x-h\right|+k\). The vertex shifts to \((h,k)\). The multiplicative factor, \(a\), causes a vertical stretch and/or a vertical flip.
In this example, \(a=-1\) and \(h=-6\) and \(k=-4\).
To get the correct daughter graph, translate the parent 6 units left and 4 units down. Since \(a<0\), the daughter points down.
The correct plot is Plot 2.
\[f(x) = - \left|{x + 6}\right| - 4\]
Question
Which plot matches the function:
\[f(x) = 5 - \left|{x - 2}\right|\]
Plot 1
Plot 2
Plot 3
Plot 4
Solution
The parent function is \(f(x)=|x|\), which has a vertex at the origin.
The general parameterization is \(f(x)=a\left|x-h\right|+k\). The vertex shifts to \((h,k)\). The multiplicative factor, \(a\), causes a vertical stretch and/or a vertical flip.
In this example, \(a=-1\) and \(h=2\) and \(k=5\).
To get the correct daughter graph, translate the parent 2 units right and 5 units up. Since \(a<0\), the daughter points down.
The correct plot is Plot 2.
\[f(x) = 5 - \left|{x - 2}\right|\]
Question
Which plot matches the function:
\[f(x) = \left|{x - 3}\right| + 1\]
Plot 1
Plot 2
Plot 3
Plot 4
Solution
The parent function is \(f(x)=|x|\), which has a vertex at the origin.
The general parameterization is \(f(x)=a\left|x-h\right|+k\). The vertex shifts to \((h,k)\). The multiplicative factor, \(a\), causes a vertical stretch and/or a vertical flip.
In this example, \(a=1\) and \(h=3\) and \(k=1\).
To get the correct daughter graph, translate the parent 3 units right and 1 units up. Since \(a>0\), the daughter points up.
The correct plot is Plot 2.
\[f(x) = \left|{x - 3}\right| + 1\]
Question
Let functions \(f\) and \(g\) be defined by the table below.
\(x\)
\(f(x)\)
\(g(x)\)
1
9
1
2
3
2
3
7
3
4
6
6
5
10
10
6
5
9
7
4
7
8
2
4
9
8
5
10
1
8
Evaluate the following:
\(f(g(3)) =\)
\(g(f(10)) =\)
\(f(f(2)) =\)
\(g(g(10)) =\)
\(f(g(g(4))) =\)
Solution
\(f(g(3))=f(3)=7\)
\(g(f(10))=g(1)=1\)
\(f(f(2))=f(3)=7\)
\(g(g(10))=g(8)=4\)
\(f(g(g(4)))=f(g(6))=f(9)=8\)
Question
Let functions \(f\) and \(g\) be defined by the table below.
\(x\)
\(f(x)\)
\(g(x)\)
1
9
9
2
7
7
3
1
3
4
6
1
5
4
2
6
3
10
7
10
4
8
5
8
9
2
6
10
8
5
Evaluate the following:
\(f(g(5)) =\)
\(g(f(10)) =\)
\(f(f(1)) =\)
\(g(g(9)) =\)
\(f(g(g(10))) =\)
Solution
\(f(g(5))=f(2)=7\)
\(g(f(10))=g(8)=8\)
\(f(f(1))=f(9)=2\)
\(g(g(9))=g(6)=10\)
\(f(g(g(10)))=f(g(5))=f(2)=7\)
Question
Let functions \(f\) and \(g\) be defined by the table below.
\(x\)
\(f(x)\)
\(g(x)\)
1
1
5
2
6
4
3
10
2
4
3
7
5
8
9
6
4
1
7
7
10
8
5
6
9
9
3
10
2
8
Evaluate the following:
\(f(g(3)) =\)
\(g(f(2)) =\)
\(f(f(10)) =\)
\(g(g(1)) =\)
\(f(g(g(8))) =\)
Solution
\(f(g(3))=f(2)=6\)
\(g(f(2))=g(6)=1\)
\(f(f(10))=f(2)=6\)
\(g(g(1))=g(5)=9\)
\(f(g(g(8)))=f(g(6))=f(1)=1\)
Question
Let functions \(f\) and \(g\) be defined by the table below.
\(x\)
\(f(x)\)
\(g(x)\)
1
4
10
2
5
7
3
9
3
4
6
9
5
8
6
6
10
1
7
3
8
8
1
2
9
2
4
10
7
5
Evaluate the following:
\(f(g(9)) =\)
\(g(f(1)) =\)
\(f(f(6)) =\)
\(g(g(7)) =\)
\(f(g(g(3))) =\)
Solution
\(f(g(9))=f(4)=6\)
\(g(f(1))=g(4)=9\)
\(f(f(6))=f(10)=7\)
\(g(g(7))=g(8)=2\)
\(f(g(g(3)))=f(g(3))=f(3)=9\)
Question
Let functions \(f\) and \(g\) be defined by the table below.
\(x\)
\(f(x)\)
\(g(x)\)
1
10
1
2
6
8
3
1
4
4
3
6
5
5
2
6
2
9
7
9
3
8
4
10
9
8
5
10
7
7
Evaluate the following:
\(f(g(3)) =\)
\(g(f(3)) =\)
\(f(f(8)) =\)
\(g(g(5)) =\)
\(f(g(g(7))) =\)
Solution
\(f(g(3))=f(4)=3\)
\(g(f(3))=g(1)=1\)
\(f(f(8))=f(4)=3\)
\(g(g(5))=g(2)=8\)
\(f(g(g(7)))=f(g(3))=f(4)=3\)
Question
Let functions \(f\) and \(g\) be defined by the table below.
\(x\)
\(f(x)\)
\(g(x)\)
1
3
8
2
5
10
3
2
6
4
7
3
5
10
5
6
9
7
7
1
2
8
8
9
9
4
4
10
6
1
Evaluate the following:
\(f(g(4)) =\)
\(g(f(5)) =\)
\(f(f(5)) =\)
\(g(g(10)) =\)
\(f(g(g(10))) =\)
Solution
\(f(g(4))=f(3)=2\)
\(g(f(5))=g(10)=1\)
\(f(f(5))=f(10)=6\)
\(g(g(10))=g(1)=8\)
\(f(g(g(10)))=f(g(1))=f(8)=8\)
Question
Let functions \(f\) and \(g\) be defined by the table below.
\(x\)
\(f(x)\)
\(g(x)\)
1
1
5
2
5
1
3
6
3
4
8
10
5
2
6
6
7
2
7
10
7
8
4
9
9
9
4
10
3
8
Evaluate the following:
\(f(g(3)) =\)
\(g(f(3)) =\)
\(f(f(8)) =\)
\(g(g(9)) =\)
\(f(g(g(7))) =\)
Solution
\(f(g(3))=f(3)=6\)
\(g(f(3))=g(6)=2\)
\(f(f(8))=f(4)=8\)
\(g(g(9))=g(4)=10\)
\(f(g(g(7)))=f(g(7))=f(7)=10\)
Question
Let functions \(f\) and \(g\) be defined by the table below.
\(x\)
\(f(x)\)
\(g(x)\)
1
6
7
2
10
1
3
8
6
4
2
10
5
5
4
6
1
8
7
7
3
8
3
5
9
4
2
10
9
9
Evaluate the following:
\(f(g(3)) =\)
\(g(f(4)) =\)
\(f(f(7)) =\)
\(g(g(8)) =\)
\(f(g(g(8))) =\)
Solution
\(f(g(3))=f(6)=1\)
\(g(f(4))=g(2)=1\)
\(f(f(7))=f(7)=7\)
\(g(g(8))=g(5)=4\)
\(f(g(g(8)))=f(g(5))=f(4)=2\)
Question
Let functions \(f\) and \(g\) be defined by the table below.
\(x\)
\(f(x)\)
\(g(x)\)
1
3
8
2
9
4
3
8
2
4
10
7
5
5
1
6
2
10
7
7
3
8
1
9
9
4
5
10
6
6
Evaluate the following:
\(f(g(3)) =\)
\(g(f(4)) =\)
\(f(f(6)) =\)
\(g(g(10)) =\)
\(f(g(g(2))) =\)
Solution
\(f(g(3))=f(2)=9\)
\(g(f(4))=g(10)=6\)
\(f(f(6))=f(2)=9\)
\(g(g(10))=g(6)=10\)
\(f(g(g(2)))=f(g(4))=f(7)=7\)
Question
Let functions \(f\) and \(g\) be defined by the table below.
\(x\)
\(f(x)\)
\(g(x)\)
1
8
5
2
1
10
3
5
8
4
7
2
5
3
1
6
6
4
7
9
7
8
10
3
9
4
9
10
2
6
Evaluate the following:
\(f(g(2)) =\)
\(g(f(7)) =\)
\(f(f(2)) =\)
\(g(g(3)) =\)
\(f(g(g(4))) =\)
Solution
\(f(g(2))=f(10)=2\)
\(g(f(7))=g(9)=9\)
\(f(f(2))=f(1)=8\)
\(g(g(3))=g(8)=3\)
\(f(g(g(4)))=f(g(2))=f(10)=2\)
Question
Let functions \(f\) and \(g\) be defined by the graph below.
Evaluate the following:
\(g(f(1)) =\)
\(f(g(8)) =\)
Solution
\(g(f(1))=g(9)=7\)
\(f(g(8))=f(9)=3\)
Question
Let functions \(f\) and \(g\) be defined by the graph below.
Evaluate the following:
\(g(f(9)) =\)
\(f(g(1)) =\)
Solution
\(g(f(9))=g(8)=3\)
\(f(g(1))=f(7)=5\)
Question
Let functions \(f\) and \(g\) be defined by the graph below.
Evaluate the following:
\(g(f(1)) =\)
\(f(g(2)) =\)
Solution
\(g(f(1))=g(6)=7\)
\(f(g(2))=f(9)=8\)
Question
Let functions \(f\) and \(g\) be defined by the graph below.
Evaluate the following:
\(g(f(1)) =\)
\(f(g(4)) =\)
Solution
\(g(f(1))=g(2)=9\)
\(f(g(4))=f(5)=5\)
Question
Let functions \(f\) and \(g\) be defined by the graph below.
Evaluate the following:
\(g(f(1)) =\)
\(f(g(1)) =\)
Solution
\(g(f(1))=g(2)=5\)
\(f(g(1))=f(7)=8\)
Question
Let functions \(f\) and \(g\) be defined by the graph below.
Evaluate the following:
\(g(f(8)) =\)
\(f(g(9)) =\)
Solution
\(g(f(8))=g(4)=6\)
\(f(g(9))=f(1)=7\)
Question
Let functions \(f\) and \(g\) be defined by the graph below.
Evaluate the following:
\(g(f(1)) =\)
\(f(g(2)) =\)
Solution
\(g(f(1))=g(7)=7\)
\(f(g(2))=f(9)=6\)
Question
Let functions \(f\) and \(g\) be defined by the graph below.
Evaluate the following:
\(g(f(5)) =\)
\(f(g(4)) =\)
Solution
\(g(f(5))=g(3)=1\)
\(f(g(4))=f(9)=8\)
Question
Let functions \(f\) and \(g\) be defined by the graph below.
Evaluate the following:
\(g(f(1)) =\)
\(f(g(3)) =\)
Solution
\(g(f(1))=g(9)=2\)
\(f(g(3))=f(9)=7\)
Question
Let functions \(f\) and \(g\) be defined by the graph below.
Evaluate the following:
\(g(f(2)) =\)
\(f(g(1)) =\)
Solution
\(g(f(2))=g(3)=3\)
\(f(g(1))=f(9)=4\)
Question
Let function \(f\) be defined by the graph below.
Over what interval is the function positive?
Over what interval is the function negative?
Over what interval is the function increasing?
Over what interval is the function decreasing?
Solution
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-5,-4)U(0,4)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-4,0)U(4,5)
The function is increasing when an ant going left-to-right is walking up hill: (-2,2)
The function is decreasing when an ant going left-to-right is walking down hill: (-5,-2)U(2,5)
Question
Let function \(f\) be defined by the graph below.
Over what interval is the function positive?
Over what interval is the function negative?
Over what interval is the function increasing?
Over what interval is the function decreasing?
Solution
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-3,1)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-5,-3)U(1,5)
The function is increasing when an ant going left-to-right is walking up hill: (-5,-1)U(3,5)
The function is decreasing when an ant going left-to-right is walking down hill: (-1,3)
Question
Let function \(f\) be defined by the graph below.
Over what interval is the function positive?
Over what interval is the function negative?
Over what interval is the function increasing?
Over what interval is the function decreasing?
Solution
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-1,3)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-5,-1)U(3,5)
The function is increasing when an ant going left-to-right is walking up hill: (-3,1)
The function is decreasing when an ant going left-to-right is walking down hill: (-5,-3)U(1,5)
Question
Let function \(f\) be defined by the graph below.
Over what interval is the function positive?
Over what interval is the function negative?
Over what interval is the function increasing?
Over what interval is the function decreasing?
Solution
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-5,-4)U(0,4)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-4,0)U(4,5)
The function is increasing when an ant going left-to-right is walking up hill: (-2,2)
The function is decreasing when an ant going left-to-right is walking down hill: (-5,-2)U(2,5)
Question
Let function \(f\) be defined by the graph below.
Over what interval is the function positive?
Over what interval is the function negative?
Over what interval is the function increasing?
Over what interval is the function decreasing?
Solution
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-1,3)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-5,-1)U(3,5)
The function is increasing when an ant going left-to-right is walking up hill: (-3,1)
The function is decreasing when an ant going left-to-right is walking down hill: (-5,-3)U(1,5)
Question
Let function \(f\) be defined by the graph below.
Over what interval is the function positive?
Over what interval is the function negative?
Over what interval is the function increasing?
Over what interval is the function decreasing?
Solution
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-5,-4)U(0,4)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-4,0)U(4,5)
The function is increasing when an ant going left-to-right is walking up hill: (-2,2)
The function is decreasing when an ant going left-to-right is walking down hill: (-5,-2)U(2,5)
Question
Let function \(f\) be defined by the graph below.
Over what interval is the function positive?
Over what interval is the function negative?
Over what interval is the function increasing?
Over what interval is the function decreasing?
Solution
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-1,3)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-5,-1)U(3,5)
The function is increasing when an ant going left-to-right is walking up hill: (-3,1)
The function is decreasing when an ant going left-to-right is walking down hill: (-5,-3)U(1,5)
Question
Let function \(f\) be defined by the graph below.
Over what interval is the function positive?
Over what interval is the function negative?
Over what interval is the function increasing?
Over what interval is the function decreasing?
Solution
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-3,1)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-5,-3)U(1,5)
The function is increasing when an ant going left-to-right is walking up hill: (-5,-1)U(3,5)
The function is decreasing when an ant going left-to-right is walking down hill: (-1,3)
Question
Let function \(f\) be defined by the graph below.
Over what interval is the function positive?
Over what interval is the function negative?
Over what interval is the function increasing?
Over what interval is the function decreasing?
Solution
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-4,0)U(4,5)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-5,-4)U(0,4)
The function is increasing when an ant going left-to-right is walking up hill: (-5,-2)U(2,5)
The function is decreasing when an ant going left-to-right is walking down hill: (-2,2)
Question
Let function \(f\) be defined by the graph below.
Over what interval is the function positive?
Over what interval is the function negative?
Over what interval is the function increasing?
Over what interval is the function decreasing?
Solution
The function “is positive” when \(f(x)>0\). The interval refers to the \(x\) values (inputs) that produce a positive output. Determine where the curve is above the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-5,-2)U(2,5)
The function “is negative” when \(f(x)<0\). The interval refers to the \(x\) values (inputs) that produce a negative output. Determine where the curve is below the \(x\) axis. Indicate “where” with the \(x\) values at the endpoints: (-2,2)
The function is increasing when an ant going left-to-right is walking up hill: (-5,-4)U(0,4)
The function is decreasing when an ant going left-to-right is walking down hill: (-4,0)U(4,5)
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
2
8
3
3
6
4
7
2
Let \(g(x)\) be defined with the equation \(g(x)=f(x+4)-3\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=f(x+4)-3\) implies a horizontal shift of left by 4 and a vertical shift of down by 3. The horizontal shift is an additive/subtractive change to the inputs. The vertical shift is an additive/subtractive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
2
5
4
6
6
4
8
3
Let \(g(x)\) be defined with the equation \(g(x)=f(x+4)+5\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=f(x+4)+5\) implies a horizontal shift of left by 4 and a vertical shift of up by 5. The horizontal shift is an additive/subtractive change to the inputs. The vertical shift is an additive/subtractive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
2
6
4
2
7
7
8
5
Let \(g(x)\) be defined with the equation \(g(x)=f(x-3)+4\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=f(x-3)+4\) implies a horizontal shift of right by 3 and a vertical shift of up by 4. The horizontal shift is an additive/subtractive change to the inputs. The vertical shift is an additive/subtractive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
2
8
4
7
5
3
6
2
Let \(g(x)\) be defined with the equation \(g(x)=f(x-5)-2\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=f(x-5)-2\) implies a horizontal shift of right by 5 and a vertical shift of down by 2. The horizontal shift is an additive/subtractive change to the inputs. The vertical shift is an additive/subtractive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
5
6
6
8
7
3
8
4
Let \(g(x)\) be defined with the equation \(g(x)=f(x+1)-3\). What are the 4 corresponding points on \(g\)? (Use same order.)
\(x\)
\(g(x)\)
Solution
\(x\)
\(g(x)\)
4
3
5
5
6
0
7
1
Notice \(g(4)~=~f(4 +1)-3~=~f(5)-3~=~6 -3 ~=~ 3\)
Notice \(g(5)~=~f(5 +1)-3~=~f(6)-3~=~8 -3 ~=~ 5\)
Notice \(g(6)~=~f(6 +1)-3~=~f(7)-3~=~3 -3 ~=~ 0\)
Notice \(g(7)~=~f(7 +1)-3~=~f(8)-3~=~4 -3 ~=~ 1\)
The “shortcut” is to recognize that the definition \(g(x)=f(x+1)-3\) implies a horizontal shift of left by 1 and a vertical shift of down by 3. The horizontal shift is an additive/subtractive change to the inputs. The vertical shift is an additive/subtractive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
3
4
5
7
6
8
7
2
Let \(g(x)\) be defined with the equation \(g(x)=f(x+5)-3\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=f(x+5)-3\) implies a horizontal shift of left by 5 and a vertical shift of down by 3. The horizontal shift is an additive/subtractive change to the inputs. The vertical shift is an additive/subtractive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
2
5
5
2
6
4
8
8
Let \(g(x)\) be defined with the equation \(g(x)=f(x+5)-2\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=f(x+5)-2\) implies a horizontal shift of left by 5 and a vertical shift of down by 2. The horizontal shift is an additive/subtractive change to the inputs. The vertical shift is an additive/subtractive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
4
6
5
2
6
8
7
7
Let \(g(x)\) be defined with the equation \(g(x)=f(x-3)+1\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=f(x-3)+1\) implies a horizontal shift of right by 3 and a vertical shift of up by 1. The horizontal shift is an additive/subtractive change to the inputs. The vertical shift is an additive/subtractive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
2
4
5
7
6
5
8
6
Let \(g(x)\) be defined with the equation \(g(x)=f(x-5)-3\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=f(x-5)-3\) implies a horizontal shift of right by 5 and a vertical shift of down by 3. The horizontal shift is an additive/subtractive change to the inputs. The vertical shift is an additive/subtractive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
3
5
4
2
6
3
8
8
Let \(g(x)\) be defined with the equation \(g(x)=f(x+5)-4\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=f(x+5)-4\) implies a horizontal shift of left by 5 and a vertical shift of down by 4. The horizontal shift is an additive/subtractive change to the inputs. The vertical shift is an additive/subtractive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
6
54
12
48
18
30
24
42
Let \(g(x)\) be defined with the equation \(g(x)=2\, f(\frac{1}{3}x)\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=2\, f(\frac{1}{3}x)\) implies a horizontal stretch of 3 and a vertical stretch of 2. The horizontal stretch is an multiplicative/divisive change to the inputs. The vertical stretch is an multiplicative/divisive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
6
54
12
60
18
36
24
42
Let \(g(x)\) be defined with the equation \(g(x)=\frac{1}{2}\,f(3x)\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=\frac{1}{2}\,f(3x)\) implies a horizontal stretch of 0.3333333 and a vertical stretch of 0.5. The horizontal stretch is an multiplicative/divisive change to the inputs. The vertical stretch is an multiplicative/divisive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
6
54
12
42
18
60
24
30
Let \(g(x)\) be defined with the equation \(g(x)=3\, f(\frac{1}{2}x)\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=3\, f(\frac{1}{2}x)\) implies a horizontal stretch of 2 and a vertical stretch of 3. The horizontal stretch is an multiplicative/divisive change to the inputs. The vertical stretch is an multiplicative/divisive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
6
42
12
54
18
30
24
36
Let \(g(x)\) be defined with the equation \(g(x)=2\, f(\frac{1}{3}x)\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=2\, f(\frac{1}{3}x)\) implies a horizontal stretch of 3 and a vertical stretch of 2. The horizontal stretch is an multiplicative/divisive change to the inputs. The vertical stretch is an multiplicative/divisive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
6
42
12
30
18
60
24
54
Let \(g(x)\) be defined with the equation \(g(x)=\frac{1}{2}\,f(3x)\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=\frac{1}{2}\,f(3x)\) implies a horizontal stretch of 0.3333333 and a vertical stretch of 0.5. The horizontal stretch is an multiplicative/divisive change to the inputs. The vertical stretch is an multiplicative/divisive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
6
60
12
30
18
54
24
48
Let \(g(x)\) be defined with the equation \(g(x)=\frac{1}{2}\,f(3x)\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=\frac{1}{2}\,f(3x)\) implies a horizontal stretch of 0.3333333 and a vertical stretch of 0.5. The horizontal stretch is an multiplicative/divisive change to the inputs. The vertical stretch is an multiplicative/divisive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
6
48
12
54
18
60
24
42
Let \(g(x)\) be defined with the equation \(g(x)=2\, f(\frac{1}{3}x)\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=2\, f(\frac{1}{3}x)\) implies a horizontal stretch of 3 and a vertical stretch of 2. The horizontal stretch is an multiplicative/divisive change to the inputs. The vertical stretch is an multiplicative/divisive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
6
30
12
36
18
42
24
48
Let \(g(x)\) be defined with the equation \(g(x)=\frac{1}{3}\,f(2x)\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=\frac{1}{3}\,f(2x)\) implies a horizontal stretch of 0.5 and a vertical stretch of 0.3333333. The horizontal stretch is an multiplicative/divisive change to the inputs. The vertical stretch is an multiplicative/divisive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
6
48
12
60
18
30
24
54
Let \(g(x)\) be defined with the equation \(g(x)=3\, f(\frac{1}{2}x)\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=3\, f(\frac{1}{2}x)\) implies a horizontal stretch of 2 and a vertical stretch of 3. The horizontal stretch is an multiplicative/divisive change to the inputs. The vertical stretch is an multiplicative/divisive change to the outputs.
Question
Let \(f(x)\) be defined by the table below:
\(x\)
\(f(x)\)
6
36
12
42
18
30
24
54
Let \(g(x)\) be defined with the equation \(g(x)=\frac{1}{2}\,f(3x)\). What are the 4 corresponding points on \(g\)? (Use same order.)
The “shortcut” is to recognize that the definition \(g(x)=\frac{1}{2}\,f(3x)\) implies a horizontal stretch of 0.3333333 and a vertical stretch of 0.5. The horizontal stretch is an multiplicative/divisive change to the inputs. The vertical stretch is an multiplicative/divisive change to the outputs.
Question
A simple piece-wise function \(f\) is plotted below.
It’s general daughter function, having translations, reflections, and stretches, can be expressed with 4 parameters: \(a\), \(b\), \(h\), and \(k\).
\[g(x) = a\, f(b(x-h))+k \]
Compared to the parent function, the daughter is stretched vertically by a factor of \(a\), stretch horizontally by a factor of \(\frac{1}{b}\), and then shifted right \(h\) and up \(k\).
Notice, a left or down shift gives a negative value for \(h\) or \(k\). Also, a negative stretch factor causes a reflection.
Each of those four parameters is randomly assigned \(\pm 2\) or \(\pm \frac{1}{2}\). Determine the four parameters of the daughter by inspecting the daughter’s graph:
\(a =\)
\(b =\)
\(h =\)
\(k =\)
Also, write the equation to define the daughter.
\(g(x) =\)\(f(\)\((x\)\()\)
Solution
The daughter is stretched vertically by a factor of 0.5, so \(a=0.5\).
The daughter is stretched horizontally by a factor of 0.5, so \(b=2\).
The parent’s origin (free end of long segment), is shifted to (2, 2), so \(h=2\) and \(k=2\).
When we put those parameters into definition of \(g\), we get:
\[g(x) = 0.5\,f(2(x-2))+2 \]
Personally, I find it most interesting that the horizontal transformations occur REVERSE PEMDAS. Most people would guess, using order of operations, that the horizontal translation by \(h\) would occur before the horizontal stretch by \(\frac{1}{b}\), but this would be wrong. You might remember that applying inverse operations in reverse PEMDAS is what we do to solve equations in Algebra.
Question
A simple piece-wise function \(f\) is plotted below.
It’s general daughter function, having translations, reflections, and stretches, can be expressed with 4 parameters: \(a\), \(b\), \(h\), and \(k\).
\[g(x) = a\, f(b(x-h))+k \]
Compared to the parent function, the daughter is stretched vertically by a factor of \(a\), stretch horizontally by a factor of \(\frac{1}{b}\), and then shifted right \(h\) and up \(k\).
Notice, a left or down shift gives a negative value for \(h\) or \(k\). Also, a negative stretch factor causes a reflection.
Each of those four parameters is randomly assigned \(\pm 2\) or \(\pm \frac{1}{2}\). Determine the four parameters of the daughter by inspecting the daughter’s graph:
\(a =\)
\(b =\)
\(h =\)
\(k =\)
Also, write the equation to define the daughter.
\(g(x) =\)\(f(\)\((x\)\()\)
Solution
The daughter is stretched vertically by a factor of 2 and reflected vertically, so \(a=-2\).
The daughter is stretched horizontally by a factor of 0.5 and reflected horizontally, so \(b=-2\).
The parent’s origin (free end of long segment), is shifted to (-0.5, -0.5), so \(h=-0.5\) and \(k=-0.5\).
When we put those parameters into definition of \(g\), we get:
\[g(x) = -2\,f(-2(x+0.5))-0.5 \]
Personally, I find it most interesting that the horizontal transformations occur REVERSE PEMDAS. Most people would guess, using order of operations, that the horizontal translation by \(h\) would occur before the horizontal stretch by \(\frac{1}{b}\), but this would be wrong. You might remember that applying inverse operations in reverse PEMDAS is what we do to solve equations in Algebra.
Question
A simple piece-wise function \(f\) is plotted below.
It’s general daughter function, having translations, reflections, and stretches, can be expressed with 4 parameters: \(a\), \(b\), \(h\), and \(k\).
\[g(x) = a\, f(b(x-h))+k \]
Compared to the parent function, the daughter is stretched vertically by a factor of \(a\), stretch horizontally by a factor of \(\frac{1}{b}\), and then shifted right \(h\) and up \(k\).
Notice, a left or down shift gives a negative value for \(h\) or \(k\). Also, a negative stretch factor causes a reflection.
Each of those four parameters is randomly assigned \(\pm 2\) or \(\pm \frac{1}{2}\). Determine the four parameters of the daughter by inspecting the daughter’s graph:
\(a =\)
\(b =\)
\(h =\)
\(k =\)
Also, write the equation to define the daughter.
\(g(x) =\)\(f(\)\((x\)\()\)
Solution
The daughter is stretched vertically by a factor of 0.5, so \(a=0.5\).
The daughter is stretched horizontally by a factor of 2, so \(b=0.5\).
The parent’s origin (free end of long segment), is shifted to (2, 0.5), so \(h=2\) and \(k=0.5\).
When we put those parameters into definition of \(g\), we get:
\[g(x) = 0.5\,f(0.5(x-2))+0.5 \]
Personally, I find it most interesting that the horizontal transformations occur REVERSE PEMDAS. Most people would guess, using order of operations, that the horizontal translation by \(h\) would occur before the horizontal stretch by \(\frac{1}{b}\), but this would be wrong. You might remember that applying inverse operations in reverse PEMDAS is what we do to solve equations in Algebra.
Question
A simple piece-wise function \(f\) is plotted below.
It’s general daughter function, having translations, reflections, and stretches, can be expressed with 4 parameters: \(a\), \(b\), \(h\), and \(k\).
\[g(x) = a\, f(b(x-h))+k \]
Compared to the parent function, the daughter is stretched vertically by a factor of \(a\), stretch horizontally by a factor of \(\frac{1}{b}\), and then shifted right \(h\) and up \(k\).
Notice, a left or down shift gives a negative value for \(h\) or \(k\). Also, a negative stretch factor causes a reflection.
Each of those four parameters is randomly assigned \(\pm 2\) or \(\pm \frac{1}{2}\). Determine the four parameters of the daughter by inspecting the daughter’s graph:
\(a =\)
\(b =\)
\(h =\)
\(k =\)
Also, write the equation to define the daughter.
\(g(x) =\)\(f(\)\((x\)\()\)
Solution
The daughter is stretched vertically by a factor of 2 and reflected vertically, so \(a=-2\).
The daughter is stretched horizontally by a factor of 2 and reflected horizontally, so \(b=-0.5\).
The parent’s origin (free end of long segment), is shifted to (-2, 0.5), so \(h=-2\) and \(k=0.5\).
When we put those parameters into definition of \(g\), we get:
\[g(x) = -2\,f(-0.5(x+2))+0.5 \]
Personally, I find it most interesting that the horizontal transformations occur REVERSE PEMDAS. Most people would guess, using order of operations, that the horizontal translation by \(h\) would occur before the horizontal stretch by \(\frac{1}{b}\), but this would be wrong. You might remember that applying inverse operations in reverse PEMDAS is what we do to solve equations in Algebra.
Question
A simple piece-wise function \(f\) is plotted below.
It’s general daughter function, having translations, reflections, and stretches, can be expressed with 4 parameters: \(a\), \(b\), \(h\), and \(k\).
\[g(x) = a\, f(b(x-h))+k \]
Compared to the parent function, the daughter is stretched vertically by a factor of \(a\), stretch horizontally by a factor of \(\frac{1}{b}\), and then shifted right \(h\) and up \(k\).
Notice, a left or down shift gives a negative value for \(h\) or \(k\). Also, a negative stretch factor causes a reflection.
Each of those four parameters is randomly assigned \(\pm 2\) or \(\pm \frac{1}{2}\). Determine the four parameters of the daughter by inspecting the daughter’s graph:
\(a =\)
\(b =\)
\(h =\)
\(k =\)
Also, write the equation to define the daughter.
\(g(x) =\)\(f(\)\((x\)\()\)
Solution
The daughter is stretched vertically by a factor of 0.5 and reflected vertically, so \(a=-0.5\).
The daughter is stretched horizontally by a factor of 2, so \(b=0.5\).
The parent’s origin (free end of long segment), is shifted to (-0.5, -0.5), so \(h=-0.5\) and \(k=-0.5\).
When we put those parameters into definition of \(g\), we get:
\[g(x) = -0.5\,f(0.5(x+0.5))-0.5 \]
Personally, I find it most interesting that the horizontal transformations occur REVERSE PEMDAS. Most people would guess, using order of operations, that the horizontal translation by \(h\) would occur before the horizontal stretch by \(\frac{1}{b}\), but this would be wrong. You might remember that applying inverse operations in reverse PEMDAS is what we do to solve equations in Algebra.
Question
A simple piece-wise function \(f\) is plotted below.
It’s general daughter function, having translations, reflections, and stretches, can be expressed with 4 parameters: \(a\), \(b\), \(h\), and \(k\).
\[g(x) = a\, f(b(x-h))+k \]
Compared to the parent function, the daughter is stretched vertically by a factor of \(a\), stretch horizontally by a factor of \(\frac{1}{b}\), and then shifted right \(h\) and up \(k\).
Notice, a left or down shift gives a negative value for \(h\) or \(k\). Also, a negative stretch factor causes a reflection.
Each of those four parameters is randomly assigned \(\pm 2\) or \(\pm \frac{1}{2}\). Determine the four parameters of the daughter by inspecting the daughter’s graph:
\(a =\)
\(b =\)
\(h =\)
\(k =\)
Also, write the equation to define the daughter.
\(g(x) =\)\(f(\)\((x\)\()\)
Solution
The daughter is stretched vertically by a factor of 2, so \(a=2\).
The daughter is stretched horizontally by a factor of 0.5 and reflected horizontally, so \(b=-2\).
The parent’s origin (free end of long segment), is shifted to (-2, 2), so \(h=-2\) and \(k=2\).
When we put those parameters into definition of \(g\), we get:
\[g(x) = 2\,f(-2(x+2))+2 \]
Personally, I find it most interesting that the horizontal transformations occur REVERSE PEMDAS. Most people would guess, using order of operations, that the horizontal translation by \(h\) would occur before the horizontal stretch by \(\frac{1}{b}\), but this would be wrong. You might remember that applying inverse operations in reverse PEMDAS is what we do to solve equations in Algebra.
Question
A simple piece-wise function \(f\) is plotted below.
It’s general daughter function, having translations, reflections, and stretches, can be expressed with 4 parameters: \(a\), \(b\), \(h\), and \(k\).
\[g(x) = a\, f(b(x-h))+k \]
Compared to the parent function, the daughter is stretched vertically by a factor of \(a\), stretch horizontally by a factor of \(\frac{1}{b}\), and then shifted right \(h\) and up \(k\).
Notice, a left or down shift gives a negative value for \(h\) or \(k\). Also, a negative stretch factor causes a reflection.
Each of those four parameters is randomly assigned \(\pm 2\) or \(\pm \frac{1}{2}\). Determine the four parameters of the daughter by inspecting the daughter’s graph:
\(a =\)
\(b =\)
\(h =\)
\(k =\)
Also, write the equation to define the daughter.
\(g(x) =\)\(f(\)\((x\)\()\)
Solution
The daughter is stretched vertically by a factor of 2, so \(a=2\).
The daughter is stretched horizontally by a factor of 2 and reflected horizontally, so \(b=-0.5\).
The parent’s origin (free end of long segment), is shifted to (2, 0.5), so \(h=2\) and \(k=0.5\).
When we put those parameters into definition of \(g\), we get:
\[g(x) = 2\,f(-0.5(x-2))+0.5 \]
Personally, I find it most interesting that the horizontal transformations occur REVERSE PEMDAS. Most people would guess, using order of operations, that the horizontal translation by \(h\) would occur before the horizontal stretch by \(\frac{1}{b}\), but this would be wrong. You might remember that applying inverse operations in reverse PEMDAS is what we do to solve equations in Algebra.
Question
A simple piece-wise function \(f\) is plotted below.
It’s general daughter function, having translations, reflections, and stretches, can be expressed with 4 parameters: \(a\), \(b\), \(h\), and \(k\).
\[g(x) = a\, f(b(x-h))+k \]
Compared to the parent function, the daughter is stretched vertically by a factor of \(a\), stretch horizontally by a factor of \(\frac{1}{b}\), and then shifted right \(h\) and up \(k\).
Notice, a left or down shift gives a negative value for \(h\) or \(k\). Also, a negative stretch factor causes a reflection.
Each of those four parameters is randomly assigned \(\pm 2\) or \(\pm \frac{1}{2}\). Determine the four parameters of the daughter by inspecting the daughter’s graph:
\(a =\)
\(b =\)
\(h =\)
\(k =\)
Also, write the equation to define the daughter.
\(g(x) =\)\(f(\)\((x\)\()\)
Solution
The daughter is stretched vertically by a factor of 0.5, so \(a=0.5\).
The daughter is stretched horizontally by a factor of 0.5 and reflected horizontally, so \(b=-2\).
The parent’s origin (free end of long segment), is shifted to (0.5, 0.5), so \(h=0.5\) and \(k=0.5\).
When we put those parameters into definition of \(g\), we get:
\[g(x) = 0.5\,f(-2(x-0.5))+0.5 \]
Personally, I find it most interesting that the horizontal transformations occur REVERSE PEMDAS. Most people would guess, using order of operations, that the horizontal translation by \(h\) would occur before the horizontal stretch by \(\frac{1}{b}\), but this would be wrong. You might remember that applying inverse operations in reverse PEMDAS is what we do to solve equations in Algebra.
Question
A simple piece-wise function \(f\) is plotted below.
It’s general daughter function, having translations, reflections, and stretches, can be expressed with 4 parameters: \(a\), \(b\), \(h\), and \(k\).
\[g(x) = a\, f(b(x-h))+k \]
Compared to the parent function, the daughter is stretched vertically by a factor of \(a\), stretch horizontally by a factor of \(\frac{1}{b}\), and then shifted right \(h\) and up \(k\).
Notice, a left or down shift gives a negative value for \(h\) or \(k\). Also, a negative stretch factor causes a reflection.
Each of those four parameters is randomly assigned \(\pm 2\) or \(\pm \frac{1}{2}\). Determine the four parameters of the daughter by inspecting the daughter’s graph:
\(a =\)
\(b =\)
\(h =\)
\(k =\)
Also, write the equation to define the daughter.
\(g(x) =\)\(f(\)\((x\)\()\)
Solution
The daughter is stretched vertically by a factor of 0.5, so \(a=0.5\).
The daughter is stretched horizontally by a factor of 0.5, so \(b=2\).
The parent’s origin (free end of long segment), is shifted to (0.5, 0.5), so \(h=0.5\) and \(k=0.5\).
When we put those parameters into definition of \(g\), we get:
\[g(x) = 0.5\,f(2(x-0.5))+0.5 \]
Personally, I find it most interesting that the horizontal transformations occur REVERSE PEMDAS. Most people would guess, using order of operations, that the horizontal translation by \(h\) would occur before the horizontal stretch by \(\frac{1}{b}\), but this would be wrong. You might remember that applying inverse operations in reverse PEMDAS is what we do to solve equations in Algebra.
Question
A simple piece-wise function \(f\) is plotted below.
It’s general daughter function, having translations, reflections, and stretches, can be expressed with 4 parameters: \(a\), \(b\), \(h\), and \(k\).
\[g(x) = a\, f(b(x-h))+k \]
Compared to the parent function, the daughter is stretched vertically by a factor of \(a\), stretch horizontally by a factor of \(\frac{1}{b}\), and then shifted right \(h\) and up \(k\).
Notice, a left or down shift gives a negative value for \(h\) or \(k\). Also, a negative stretch factor causes a reflection.
Each of those four parameters is randomly assigned \(\pm 2\) or \(\pm \frac{1}{2}\). Determine the four parameters of the daughter by inspecting the daughter’s graph:
\(a =\)
\(b =\)
\(h =\)
\(k =\)
Also, write the equation to define the daughter.
\(g(x) =\)\(f(\)\((x\)\()\)
Solution
The daughter is stretched vertically by a factor of 2, so \(a=2\).
The daughter is stretched horizontally by a factor of 0.5 and reflected horizontally, so \(b=-2\).
The parent’s origin (free end of long segment), is shifted to (-0.5, -0.5), so \(h=-0.5\) and \(k=-0.5\).
When we put those parameters into definition of \(g\), we get:
\[g(x) = 2\,f(-2(x+0.5))-0.5 \]
Personally, I find it most interesting that the horizontal transformations occur REVERSE PEMDAS. Most people would guess, using order of operations, that the horizontal translation by \(h\) would occur before the horizontal stretch by \(\frac{1}{b}\), but this would be wrong. You might remember that applying inverse operations in reverse PEMDAS is what we do to solve equations in Algebra.